We can assume that \(U\)  is connected, by arguing separately
for each connected component of \(U\) .
Next, by equivariant resolution of singularities (using that \(k\) 
has characteristic zero), we can assume that \(X\) 
(as well as \(U\) ) is smooth over \(k\) , while still having a \(T\) -action
[1]}.
We first show that the inclusion \(Y\rightarrow U\)  induces an isomorphism
in the derived category of motives \(DM(k)\) , \(M(Y)\rightarrow M(U)\) .
Namely, since \(X\)  is smooth over \(k\) , we have the Białynicki-Birula
decomposition, as follows. The fixed point set \(X^T\) 
is smooth over \(k\) . Write \(Z_1,\ldots ,Z_m\)  for the connected
components of \(X^T\) .
For each \(i\) , let \(Z_i^+=\lbrace x\in X: \lim _{t\rightarrow 0}tx\in Z_i\rbrace \) 
and \(Z_i^-=\lbrace x\in X: \lim _{t\rightarrow \infty }tx\in Z_i\rbrace \)  be the stable
and unstable manifolds of \(Z_i\) . Then the action of \(T\)  gives
morphisms \(Z_i^+\rightarrow Z_i\)  and \(Z_i^-\rightarrow Z_i\)  which are affine-space
bundles [2]}.
Karpenko showed that this geometric decomposition
gives a direct-sum
decomposition of Chow motives over \(k\)  [3]},
[4]}:
\(M(X)\cong \oplus _{i=1}^m M(Z_i)\lbrace a_i\rbrace ,\) 
where \(a_i:=\dim (Z_i^+)-\dim (Z_i)\) . (Here \(\text{\bf Z}\lbrace 1\rbrace \)  denotes the Lefschetz
motive, with \(M(\text{\bf P}^1)=\text{\bf Z}\lbrace 0\rbrace \oplus \text{\bf Z}\lbrace 1\rbrace \) .)
This implies another decomposition \(M(X)\cong \oplus _{i=1}^m M(Z_i)\lbrace b_i\rbrace \) ,
where \(b_i:=\dim (Z_i^-)-\dim (Z_i)\) , by inverting the \(T\) -action on \(X\) .
Here
\(a_i+\dim (Z_i)+b_i=n,\) 
by considering the action on \(T\) 
on the tangent space to \(X\)  at a point of \(Z_i\) .
The category of Chow motives is a full subcategory
of the derived category of motives, \(DM(k)\) : the thick subcategory
generated by smooth projective schemes over \(k\)  tensored with
\(\text{\bf Z}\lbrace a\rbrace =\text{\bf Z}(a)[2a]\) 
for integers \(a\)  [5]}.
Every scheme \(X\)  of finite type over \(k\) 
has a motive \(M(X)\)  and a compactly supported motive \(M^c(X)\) 
in \(DM(k)\) . We can assume that \(Z_1,\ldots ,Z_m\)  are ordered
in such a way that the closure of \(Z_i^-\)  is contained
in \(X_i:=\cup _{j\le i}Z_j^-\) . Karpenko's argument shows that
the exact triangle
\(M^c(X_{i-1})\rightarrow M^c(X_i)\rightarrow M^c(Z_i^-)\cong M(Z_i)\lbrace b_i\rbrace \) 
in \(DM(k)\)  is split [3]}.
(Indeed, his splitting on Chow groups is defined by an element
of \(CH_{\dim (Z_i)+b_i}(Z_i\times X_i)\) , and that is precisely
\(\operatorname{Hom}(M(Z_i)\lbrace b_i\rbrace ,M^c(X_i))\)  since \(Z_i\)  is smooth and proper over \(k\) .)
In particular, it follows that
\(M(X_j)\cong \oplus _{i=1}^j M(Z_i)\lbrace b_i\rbrace \) 
for each \(1\le j\le m\) , and its open complement
\(X-X_j\)  satisfies
\(M^c(X-X_j)\cong \oplus _{i=j+1}^m M(Z_i)\lbrace b_i\rbrace \) 
(Here \(X_j\)  need not be smooth,
but it is proper over \(k\) , and so its motive \(M(X_j)\) 
is the same as its compactly supported motive \(M^c(X_j)\) .)
In the notation of Conjecture REF ,
we can assume that the closed subset \(Y\)  of \(X\)  is equal to \(X_r\) 
for some \(r\le m\) .
So \(M(Y)\cong \oplus _{i=1}^r M(Z_i)\lbrace b_i\rbrace \) .
Likewise, the open subset \(U\)  is the union of the subsets
\(Z_i^+\)  with \(i\le r\) . By the splitting in \(DM(k)\)  above, applied
to the inverse action of \(T\)  on \(X\) ,
we have
\(M^c(U)\cong \oplus _{i=1}^r M(Z_i)\lbrace a_i\rbrace .\) 
Since \(U\)  is smooth of dimension \(n\)  over \(k\) , it follows
that
\(M(U)&\cong M^c(U)^*\lbrace n\rbrace \\&\cong \oplus _{i=1}^r M(Z_i)^*\lbrace n-a_i\rbrace \\&\cong \oplus _{i=1}^r M(Z_i)\lbrace n-a_i-\dim (Z_i)\rbrace \\&\cong \oplus _{i=1}^r M(Z_i)\lbrace b_i\rbrace .\) 
Thus \(M(Y)\)  is isomorphic to \(M(U)\)  in \(DM(k)\) . More precisely,
the inclusion \(Y\rightarrow U\)  induces an isomorphism \(M(Y)\rightarrow M(U)\) .
To see this, one checks from Karpenko's construction
of the splittings that for \(i,j\in \lbrace 1,\ldots ,r\rbrace \) ,
the composition \(M(Z_i)\lbrace b_i\rbrace \rightarrow M(Y)\rightarrow M(U)\rightarrow M(Z_j)\lbrace b_j\rbrace \) 
is the identity for \(i=j\)  and zero if \(i<j\) .
The schemes \(Y,U,X\)  with \(T\) -action are defined over
some finitely generated subfield of \(k\) .
So we can assume
that the field \(k\)  is finitely generated over \(\text{\bf Q}\) .
Apply Theorem REF 
to the cofiber \(A=\Sigma ^{\infty }(U/Y)\)  in \(SH(k)\) .
We showed above that the motive of \(A\)  in \(DM(k)\)  is zero.
Also, for every real embedding of \(k\) , the real realization
of \(A\)  is zero in the stable homotopy category, by Theorem
REF . (It may be that \(k\)  has no real embedding.)
Therefore, \(A=0\)  in \(SH(k)\) .
That is, the inclusion \(Y\rightarrow U\)  induces an isomorphism
in \(SH(k)\) .
Again using that \(k\)  has characteristic
zero, Bachmann showed that the \(\text{\bf P}^1\) -suspension functor
\(Q=\Sigma ^{\infty }_{\text{\bf P}^1}\colon H(k)_* \rightarrow SH(k)\) 
is conservative on \(\mathbf {A}^1\) -simply connected spaces which can be written as homotopy colimits
of spaces \(X_+\wedge \mathbf {G}_m\)  with \(X\in \mathrm {Sm}_k\) 
[7]}.
The \(\mathbf {S}^2\) -suspension of every space in \(H(k)_*\) 
is \(\mathbf {A}^1\) -simply connected. The map \(\mathbf {S}^2\wedge \mathbf {G}_m\wedge Y_+\rightarrow \mathbf {S}^2\wedge \mathbf {G}_m\wedge U_+\) 
is therefore a pointed \(\mathbf {A}^1\) -homotopy equivalence.
